Suyash Narayan Mishra, Piyush Kumar Tripathi & Alok Agrawal
|
|
- Scott Preston
- 5 years ago
- Views:
Transcription
1 IOSR Journal o Mahemaics IOSR-JM e-issn: ISSN: X. Volume Issue Ver. VI Mar - Ar. 5 PP A auberian heorem or C α β- Convergence o Cesaro Means o Orer o Funcions Suash Naraan Mishra Piush Kumar riahi & Alo Agrawal snmisra@lo. ami. eu riai@lo. ami. eu aagrawal@lo. ami. eu 3 Ami School o Alie Sciences Ami niversi ar Praesh Lucnow Camus Near Malhaur Railwa Saion Gomi Nagar Lucnow. P. Inia Absrac: he objecive o his aer o generalize cerain auberian resuls rove b Gehring [3] or summabili C ; α o sequences o uncions. In [] A. V. Bo generalize he auberian heorem or α convergence o Cesa ro means o sequences. In his aer we obain some auberian heorems or C α β convergence o Cesa ro means o orer o uncions an invesigae some o is roeries. Kewors: auberian heorem Absolue an Cesàro summabili Lebesgue Inegral Convergence. I. Inroucion he noaion is similar ha are in [3]wih he ollowing aiional einiions: I > hen A n B n enoe he n-h Cesa ro sums o orer or he series n= a n n= b n where b n = na n. A n B n enoe he a n b n. Summabili C ; α o a n will b C ; α o a n. Mishra an Srivasava [6] inrouce he Summabili meho C or uncions b generalizing C summabili meho. In his aer we iscuss some auberian heorems or C α β convergence o Cesa ro means o orer o uncions an invesigae some o is roeries. II. Deiniions an Some Preliminaries We woul lie o irs inrouce Summabili meho. Summabili meho is more general han ha o orinar convergence. I we are given a sequence s n we can consruc a generalize sequence σ n he arihmeic mean o s n b his sequence s n. I σ n is convergen in orinar sense or all n > hen we sa ha s n is summable C o he sum s. his C is calle Cesaro mean o irs orer. I s n s σ n = s +s +.+s n s ie i a sequence is convergen i is summable b meho o n + arihmeic mean. Also a series is no convergen bu is summable o he sum. he sace o summable sequences is larger han sace o convergen sequences. I σ n s as n hen we sa ha sequence s n is summable b meho o arihmeic mean. For eamle : Consier he series n= u n = u + u +.. An le σ n = s +s +.+s n I ma haen ha whereas iverges he quaniies he arihmeic mean n + o arial sum o series converges o a einie limi as n. For eamle iverges bu in his case s = s = = s = + = s 3 = s n =... Since s n = + n σ n = s +s +.+s n n + = n /n + = n+ + + n + erms /n + DOI:.979/ Page
2 A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions = + + n I n is even hen σ 4n + n = + as n an i n is o hen σ n+ n =. So in eiher case lim n σ n = s n C bu s n ε S. hereore sace o summable sequences is larger han har o sace o convergen sequences. Le be an uncion which is Lebesgue-measurable an ha : [ + R an inegrable in or an inie an which is boune in some righ han neighbourhoo o origin. Inegrals o he orm hroughou o be aen as Le. eiss an i I or lim being a Lebesgue inegral. he inegral g g. g s as we sa ha uncion is summable D o he sum s an we s D as. We noe ha or an ie i is necessar an suicien or convergence o. ha wrie he shoul converge.. C ransorm o which we enoe b is given b.3 I his eiss or an ens o a limi s as we sa ha is summable C o s an we wrie s C. We also wrie.4 we sa ha he uncion are is summable D C i his eiss an ens o a limi s as o s. When D C an D C enoe he same meho. Here we give some Gehrings generalize auberian heorems. heorem.: Suose ha α an ha is summable A α o s hen is C α β convergen o s i an onl i he uncion αβ is C α β convergen o. heorem.: Suose ha α an ha is C α β convergen. I he uncion αβ is C α β convergen o hen is summable C α o is sum or ever >. III. Now we shall rove he ollowing heorem heorem 3.: Suose ha α an ha is summable A α o s. hen or r is summable C r α o s i an onl i he uncion α β is C α β o. Proo : Necessar Coniion: I r = he heorem immeiae ollows rom he summabili o C α. I r > hen b consisenc heorem or C r α summabili Gehring [3heorem 4..] i ollows ha boh he uncions an αβ are C α β convergen o s. B Har [ Equaion 6..6] S n n r = S r+ + DOI:.979/ Page
3 r+ A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions α β an he resul ollows since a linear combinaion o uncions summable C α o isel. he suicien coniions o rove he heorem are : I r > i ma be shown as in Szasz [ 4 ] ha + αβ + n r + = u r αβ u u 3. Where αβ u = u = Case a : α = r > is obvious. Case b : α r > uing g = + αβ + n. We ge rom 3. ha g = r + αβ v v n v Where αβ u now has boune C α β- variaion over. Le N V = = r + v αβ v v N αβ r αβ r αβ v v α. α α α. hen b heorem o [5] we have V r + M v r v = M.. Where M = V α αβ :. hus αβ has boune C α β- variaion over. I is reail seen rom Minowsi s inequali ha he sum o wo C α β convergen sequences is also C α β convergen an we hereore euce ha is C α β convergen o s. Case c r=-when α = he resul reuces o auber s original heorem; when α i ollows rom above heorem. For α = he resul was rove b Hslo []. heorem 3. : Le α > γ β > γ β an suose ha a is summable C γ β o s an ha converges. hen a is summable D C α β o s. We irs rove his heorem uner unreasonable einiion.. However i he resul hols wih. hen i mus also hol uner he einiion o.3. his ollows rom he ollowing Lemmas. Lemma 3.: Le. Suose ha L or inie C accoring o he einiion.3..suose ha Deine or 3. or Le enoe he eression corresoning o bu wih relace b. hen. 3.3 hus is summable C uner he einiion.3. DOI:.979/ Page
4 A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions Lemma 3.: Le he hohesis be as in Lemma 3.an eine as above. Le an.hen D C summabili o an are equivalen. Proo o Lemma 3.: We are given ha or some > 3.3 Bu since i 3.3 hols or given i hols or an greaer i mus hol or all suicienl large. Now b sanar roeries o racional inegrals an since we have u u u u 3.4 Since 3.3 hols his will ollow rom Minowsi s inequali i we rove ha 3.5 Now i ollows a once rom he einiion ha or I hen or we have so ha Cons. = b 3.4. Proo o Lemma 3.: We use noaions as in Lemma 3. an wrie urher or he eression corresoning o bu wih relace b. DOI:.979/ Page
5 A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions DOI:.979/ Page We now ha or an ie convergence o is equivalen o he convergence o.hen he conclusion will ollow rom Minowsi s inequali i we show ha 3.6 where we ae 3.6 as incluing he asserion ha he inegral eine b converges or all. For large we have 3.7 Hence he convergence o ollows a once b a resul ue o []. Now 3.6 is equivalen o c. 3.8 Le be an suicienl large consan. hen 3.8 will ollow rom Minowsi s inequali i we show ha c. 3.9 c. 3.
6 A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions B 3.9 we have c = O O. Hence 3.9 ollows. B 3.7 he eression on he le o 3. oes no ecee a consan. hus c o 3. B an obvious change o variables he eression 3. is equal o o o C C. he resul ollows. Proo o heorem 3. : We ivie he roo ino he ollowing cases. Case I. Case II. Case III. Here we observe ha Case I an II ollow rom case III wih he ai o heorem 3.. ' For i Choose an summabili C imlies summabili ' C b heorem 3. an i ollows rom Case III ha his imlies D C. Hence i is suicien o consier he case III onl. ' Proo o Case III : Since s C imlies ha s C or ' o here is no loss o generali in consiering he Case is a osiive ineger. We have C 3. DOI:.979/ Page
7 A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions Now b einiion. Puing = an we see ha. 3.3 We also wrie. R I is clear ha whenever converges R. I ollows immeiael rom 3.3 ha is eine or > an ha R as R o an hence ha or o 3.4 Inegraing 3.4 b ars imeswe euce wih he hel o 3.3 ha C. 3.5 I is veriie ha eression in 3.6 is o. 3.6 Le R. In ac or ie we have uniorml in R. 3.7 his ma be rove b inucion on i we have DOI:.979/ Page
8 A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions R = hence he resul is evien. Suose ha an assume he resul rue or. Inegraing b ars we have R. he irs erm is o require orer b 3.7 wih relace b - an he secon b inucion hohesis. Now inegraing 3.6 b ars we have = R = R. Since he inegrae erm ens o as is boune an R as. sing 3.7 an uing we see ha he eression in curl braces C C C Again using 3.8 he inner inegral C 3.8 on uing he eression on he righ o 3.9 is equal o C C Since he inegral converges. Hence he resul ollows. Reerences []. A.V. bo Some heorems on Summabili 95. []. J. M.Hslo A auberian heorem or absolue summabilij.lonon Mah.Soc.Vol [3]. F.W. Gehring A su o α variaion I rans. Amer. Mah.Soc.vol [4]. O Szasz On roucs o summabili mehosproc.amer.mah.soc.vol [5]. G. H. Har J. E.Lilewooan Pola Inequaliies934. [6]. Mishra B. P. an Srivasava A.P. Some remars on absolue Summabili o uncions base on C mehos.o aear in Jour. Na. Aca. o Mah. Summabili DOI:.979/ Page
Chapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
More informationTopics in Combinatorial Optimization May 11, Lecture 22
8.997 Topics in Combinaorial Opimizaion May, 004 Lecure Lecurer: Michel X. Goemans Scribe: Alanha Newman Muliflows an Disjoin Pahs Le G = (V,E) be a graph an le s,,s,,...s, V be erminals. Our goal is o
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More informationAn introduction to evolution PDEs November 16, 2018 CHAPTER 5 - MARKOV SEMIGROUP
An inroucion o evoluion PDEs November 6, 8 CHAPTER 5 - MARKOV SEMIGROUP Conens. Markov semigroup. Asympoic of Markov semigroups 3.. Srong posiiviy coniion an Doeblin Theorem 3.. Geomeric sabiliy uner Harris
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationChapter 5. Localization. 5.1 Localization of categories
Chaper 5 Localizaion Consider a caegory C and a amily o morphisms in C. The aim o localizaion is o ind a new caegory C and a uncor Q : C C which sends he morphisms belonging o o isomorphisms in C, (Q,
More informationTheory of! Partial Differential Equations-I!
hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-I! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationLecture #6: Continuous-Time Signals
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Lecure #6: Coninuous-Time Signals. Inroducion In his lecure, we discussed he ollowing opics:. Mahemaical represenaion and ransormaions
More informationSUFFICIENT CONDITIONS FOR EXISTENCE SOLUTION OF LINEAR TWO-POINT BOUNDARY PROBLEM IN MINIMIZATION OF QUADRATIC FUNCTIONAL
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, Series A, OF HE ROMANIAN ACADEMY Volume, Number 4/200, pp 287 293 SUFFICIEN CONDIIONS FOR EXISENCE SOLUION OF LINEAR WO-POIN BOUNDARY PROBLEM IN
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationSOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, SABAH IFTIKHAR AND FARHAT SAFDAR
Inernaional Journal o Analysis and Applicaions Volume 16, Number 3 2018, 427-436 URL: hps://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-427 SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
More informationTheory of! Partial Differential Equations!
hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationIntegral representations and new generating functions of Chebyshev polynomials
Inegral represenaions an new generaing funcions of Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, 39 186 Roma, Ialy email:
More informationarxiv: v1 [math.gm] 7 Nov 2017
A TOUR ON THE MASTER FUNCTION THEOPHILUS AGAMA arxiv:7.0665v [mah.gm] 7 Nov 07 Absrac. In his aer we sudy a funcion defined on naural numbers having eacly wo rime facors. Using his funcion, we esablish
More informationOn a problem of Graham By E. ERDŐS and E. SZEMERÉDI (Budapest) GRAHAM stated the following conjecture : Let p be a prime and a 1,..., ap p non-zero re
On a roblem of Graham By E. ERDŐS and E. SZEMERÉDI (Budaes) GRAHAM saed he following conjecure : Le be a rime and a 1,..., a non-zero residues (mod ). Assume ha if ' a i a i, ei=0 or 1 (no all e i=0) is
More informationA note to the convergence rates in precise asymptotics
He Journal of Inequaliies and Alicaions 203, 203:378 h://www.journalofinequaliiesandalicaions.com/conen/203//378 R E S E A R C H Oen Access A noe o he convergence raes in recise asymoics Jianjun He * *
More informationThe Fundamental Theorems of Calculus
FunamenalTheorems.nb 1 The Funamenal Theorems of Calculus You have now been inrouce o he wo main branches of calculus: ifferenial calculus (which we inrouce wih he angen line problem) an inegral calculus
More informationConvolution. Lecture #6 2CT.3 8. BME 333 Biomedical Signals and Systems - J.Schesser
Convoluion Lecure #6 C.3 8 Deiniion When we compue he ollowing inegral or τ and τ we say ha he we are convoluing wih g d his says: ae τ, lip i convolve in ime -τ, hen displace i in ime by seconds -τ, and
More informationChapter 10. Optimization: More than One Choice Variable
Chaper Opimiaion: More han One Choice Variable William Sanle Jevons 85-88 Carl Menger 8 9. Opimiaion Problems Chaper 9: ma u one choice variable: consumpion Chaper : ma u o choice variables: leisure Chaper
More informationInternational Journal of Mathematical Archive-3(2), 2012, Page: Available online through ISSN
Inernaional Jornal o Mahemaical Archive- age: 59-57 Available online hrogh wwwijmaino ISSN 9 546 A NON-OA OUNDARY VAUE ROEM WIH INEGRA ONDIIONS OR A OURH ORDER SEUDOHYEROI EQUAION Azizbayov EI* an Y Mehraliyev
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationDirac s hole theory and the Pauli principle: clearing up the confusion.
Dirac s hole heory and he Pauli rincile: clearing u he conusion. Dan Solomon Rauland-Borg Cororaion 8 W. Cenral Road Moun Prosec IL 656 USA Email: dan.solomon@rauland.com Absrac. In Dirac s hole heory
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationarxiv: v3 [math.ca] 14 May 2009
SINGULAR OSCILLATORY INTEGRALS ON R n arxiv:77.757v3 [mah.ca] 4 May 9 M. PAPADIMITRAKIS AND I. R. PARISSIS Absrac. Le P,n enoe he space of all real polynomials of egree a mos on R n. We prove a new esimae
More informationThe Natural Logarithm
The Naural Logarihm 5-4-007 The Power Rule says n = n + n+ + C provie ha n. The formula oes no apply o. An anierivaive F( of woul have o saisfy F( =. Bu he Funamenal Theorem implies ha if > 0, hen Thus,
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationLecture 6 - Testing Restrictions on the Disturbance Process (References Sections 2.7 and 2.10, Hayashi)
Lecure 6 - esing Resricions on he Disurbance Process (References Secions 2.7 an 2.0, Hayashi) We have eveloe sufficien coniions for he consisency an asymoic normaliy of he OLS esimaor ha allow for coniionally
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationBernoulli numbers. Francesco Chiatti, Matteo Pintonello. December 5, 2016
UNIVERSITÁ DEGLI STUDI DI PADOVA, DIPARTIMENTO DI MATEMATICA TULLIO LEVI-CIVITA Bernoulli numbers Francesco Chiai, Maeo Pinonello December 5, 206 During las lessons we have proved he Las Ferma Theorem
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationOn Two Integrability Methods of Improper Integrals
Inernaional Journal of Mahemaics and Compuer Science, 13(218), no. 1, 45 5 M CS On Two Inegrabiliy Mehods of Improper Inegrals H. N. ÖZGEN Mahemaics Deparmen Faculy of Educaion Mersin Universiy, TR-33169
More informationSections 2.2 & 2.3 Limit of a Function and Limit Laws
Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationChapter 2 The Derivative Applied Calculus 107. We ll need a rule for finding the derivative of a product so we don t have to multiply everything out.
Chaper The Derivaive Applie Calculus 107 Secion 4: Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Eample
More informationVariational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial
More informationTHE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationThe Miki-type identity for the Apostol-Bernoulli numbers
Annales Mahemaicae e Informaicae 46 6 pp. 97 4 hp://ami.ef.hu The Mii-ype ideniy for he Aposol-Bernoulli numbers Orli Herscovici, Toufi Mansour Deparmen of Mahemaics, Universiy of Haifa, 3498838 Haifa,
More informationOnline Learning with Partial Feedback. 1 Online Mirror Descent with Estimated Gradient
Avance Course in Machine Learning Spring 2010 Online Learning wih Parial Feeback Hanous are joinly prepare by Shie Mannor an Shai Shalev-Shwarz In previous lecures we alke abou he general framework of
More informationD.I. Survival models and copulas
D- D. SURVIVAL COPULA D.I. Survival moels an copulas Definiions, relaionships wih mulivariae survival isribuion funcions an relaionships beween copulas an survival copulas. D.II. Fraily moels Use of a
More informationLecture 16 (Momentum and Impulse, Collisions and Conservation of Momentum) Physics Spring 2017 Douglas Fields
Lecure 16 (Momenum and Impulse, Collisions and Conservaion o Momenum) Physics 160-02 Spring 2017 Douglas Fields Newon s Laws & Energy The work-energy heorem is relaed o Newon s 2 nd Law W KE 1 2 1 2 F
More informationAQA Maths M2. Topic Questions from Papers. Differential Equations. Answers
AQA Mahs M Topic Quesions from Papers Differenial Equaions Answers PhysicsAndMahsTuor.com Q Soluion Marks Toal Commens M 600 0 = A Applying Newonís second law wih 0 and. Correc equaion = 0 dm Separaing
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationREPRESENTATION AND GAUSSIAN BOUNDS FOR THE DENSITY OF BROWNIAN MOTION WITH RANDOM DRIFT
Communicaions on Sochasic Analysis Vol. 1, No. 2 (216) 151-162 Serials Publicaions www.serialspublicaions.com REPRESENTATION AND GAUSSIAN BOUNDS FOR THE DENSITY OF BROWNIAN MOTION WITH RANDOM DRIFT AZMI
More informationCheck in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2
7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationMath 106: Review for Final Exam, Part II. (x x 0 ) 2 = !
Mah 6: Review for Final Exam, Par II. Use a second-degree Taylor polynomial o esimae 8. We choose f(x) x and x 7 because 7 is he perfec cube closes o 8. f(x) x / f(7) f (x) x / f (7) x / 7 / 7 f (x) 9
More informationAdvanced Integration Techniques: Integration by Parts We may differentiate the product of two functions by using the product rule:
Avance Inegraion Techniques: Inegraion by Pars We may iffereniae he prouc of wo funcions by using he prouc rule: x f(x)g(x) = f (x)g(x) + f(x)g (x). Unforunaely, fining an anierivaive of a prouc is no
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
More informationScientific Research of the Institute of Mathematics and Computer Science DIFFERENT VARIANTS OF THE BOUNDARY ELEMENT METHOD FOR PARABOLIC EQUATIONS
Scieniic Research o he Insiue o Mahemaics and Compuer Science DIERENT VARIANTS O THE BOUNDARY ELEMENT METHOD OR PARABOLIC EQUATIONS Ewa Majchrzak,, Ewa Ładyga Jerzy Mendakiewicz, Alicja Piasecka Belkhaya
More informationSterilization D Values
Seriliaion D Values Seriliaion by seam consis of he simple observaion ha baceria die over ime during exposure o hea. They do no all live for a finie period of hea exposure and hen suddenly die a once,
More informationBoundary Control of the Viscous Generalized Camassa-Holm Equation
ISSN 1749-3889 prin, 1749-3897 online Inernaional Journal of Nonlinear Science Vol89 No,pp173-181 Bounary Conrol of he Viscous Generalize Camassa-Holm Equaion Yiping Meng 1,, Lixin Tian 1 School of Mahemaics
More informationA Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero
JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS 24, 7887 1997 ARICLE NO. AY965143 A Necessary and Sufficien Condiion for he Soluions of a Funcional Differenial Equaion o Be Oscillaory or end o Zero Piambar
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationF This leads to an unstable mode which is not observable at the output thus cannot be controlled by feeding back.
Lecure 8 Las ime: Semi-free configuraion design This is equivalen o: Noe ns, ener he sysem a he same place. is fixed. We design C (and perhaps B. We mus sabilize if i is given as unsable. Cs ( H( s = +
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationCharacteristics of Linear System
Characerisics o Linear Sysem h g h : Impulse response F G : Frequency ranser uncion Represenaion o Sysem in ime an requency. Low-pass iler g h G F he requency ranser uncion is he Fourier ransorm o he impulse
More informationMath 221: Mathematical Notation
Mah 221: Mahemaical Noaion Purpose: One goal in any course is o properly use he language o ha subjec. These noaions summarize some o he major conceps and more diicul opics o he uni. Typing hem helps you
More informationOn Hankel type transform of generalized Mathieu series
Inernaional Journal of Saisika and Mahemaika ISSN: 77-79 E-ISSN: 49-865 Volume Issue 3 3- On Hankel ye ransform of generalized Mahieu series BBWahare MAEER s MIT ACSC Alandi Pune-45 Maharashra India Corresondence
More informationPractice Problems - Week #4 Higher-Order DEs, Applications Solutions
Pracice Probles - Wee #4 Higher-Orer DEs, Applicaions Soluions 1. Solve he iniial value proble where y y = 0, y0 = 0, y 0 = 1, an y 0 =. r r = rr 1 = rr 1r + 1, so he general soluion is C 1 + C e x + C
More informationMapping Properties Of The General Integral Operator On The Classes R k (ρ, b) And V k (ρ, b)
Applied Mahemaics E-Noes, 15(215), 14-21 c ISSN 167-251 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Mapping Properies Of The General Inegral Operaor On The Classes R k (ρ, b) And V k
More informationSoviet Rail Network, 1955
7.1 Nework Flow Sovie Rail Nework, 19 Reerence: On he hiory o he ranporaion and maximum low problem. lexander Schrijver in Mah Programming, 91: 3, 00. (See Exernal Link ) Maximum Flow and Minimum Cu Max
More informationBoundedness and Stability of Solutions of Some Nonlinear Differential Equations of the Third-Order.
Boundedness Sabili of Soluions of Some Nonlinear Differenial Equaions of he Third-Order. A.T. Ademola, M.Sc. * P.O. Arawomo, Ph.D. Deparmen of Mahemaics Saisics, Bowen Universi, Iwo, Nigeria. Deparmen
More informationTHE STORY OF LANDEN, THE HYPERBOLA AND THE ELLIPSE
THE STORY OF LANDEN, THE HYPERBOLA AND THE ELLIPSE VICTOR H. MOLL, JUDITH L. NOWALSKY, AND LEONARDO SOLANILLA Absrac. We esablish a relaion among he arc lenghs of a hyperbola, a circle and an ellipse..
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationChapter 3 Common Families of Distributions
Chaer 3 Common Families of Disribuions Secion 31 - Inroducion Purose of his Chaer: Caalog many of common saisical disribuions (families of disribuions ha are indeed by one or more arameers) Wha we should
More informationHow to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
More informationarxiv:math/ v1 [math.nt] 3 Nov 2005
arxiv:mah/0511092v1 [mah.nt] 3 Nov 2005 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON AND S. M. GONEK Absrac. Le πs denoe he argumen of he Riemann zea-funcion a he poin 1 + i. Assuming
More informationL p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity
ANNALES POLONICI MATHEMATICI LIV.2 99) L p -L q -Time decay esimae for soluion of he Cauchy problem for hyperbolic parial differenial equaions of linear hermoelasiciy by Jerzy Gawinecki Warszawa) Absrac.
More informationOn the Stability of the n-dimensional Quadratic and Additive Functional Equation in Random Normed Spaces via Fixed Point Method
In. Journal of Mah. Analysis, Vol. 7, 013, no. 49, 413-48 HIKARI Ld, www.m-hikari.com hp://d.doi.org/10.1988/ijma.013.36165 On he Sabiliy of he n-dimensional Quadraic and Addiive Funcional Equaion in Random
More informationPrakash Chandra Rautaray 1, Ellipse 2
Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: 48-96 www.ijera.com Vol. 3, Issue, Jauar -Februar 3,.36-337 Degree Of Aroimaio Of Fucios B Modified Parial
More informationy h h y
Porland Communiy College MTH 51 Lab Manual Limis and Coninuiy Aciviy 4 While working problem 3.6 you compleed Table 4.1 (ormerly Table 3.1). In he cone o ha problem he dierence quoien being evaluaed reurned
More informationFaα-Irresolute Mappings
BULLETIN o he Bull. Malaysian Mah. Sc. Soc. (Second Series) 24 (2001) 193-199 MLYSIN MTHEMTICL SCIENCES SOCIETY Faα-Irresolue Mappings 1 R.K. SRF, 2 M. CLDS ND 3 SEEM MISHR 1 Deparmen o Mahemaics, Governmen
More informationOn Likelihood Ratio and Stochastic Order. for Skew-symmetric Distributions. with a Common Kernel
In. J. Conemp. Mah. Sciences Vol. 8 3 no. 957-967 HIKARI Ld www.m-hikari.com hp://d.doi.org/.988/icms.3.38 On Likelihood Raio and Sochasic Order or Skew-symmeric Disribuions wih a Common Kernel Werner
More informationA NOTE ON S(t) AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION
Bull. London Mah. Soc. 39 2007 482 486 C 2007 London Mahemaical Sociey doi:10.1112/blms/bdm032 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON and S. M. GONEK Absrac Le πs denoe he
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationSettling Time Design and Parameter Tuning Methods for Finite-Time P-PI Control
Journal of Conrol Science an Engineering (6) - oi:.765/8-/6.. D DAVID PUBLISHING Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol Keigo Hiruma, Hisaazu Naamura an Yasuyui Saoh. Deparmen
More informationPROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES
PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationOn Customized Goods, Standard Goods, and Competition
On Cusomize Goos, Sanar Goos, an Compeiion Nilari. Syam C. T. auer College of usiness Universiy of Houson 85 Melcher Hall, Houson, TX 7704 Email: nbsyam@uh.eu Phone: (71 74 4568 Fax: (71 74 457 Nana Kumar
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More informationTHE SINE INTEGRAL. x dt t
THE SINE INTEGRAL As one learns in elemenary calculus, he limi of sin(/ as vanishes is uniy. Furhermore he funcion is even and has an infinie number of zeros locaed a ±n for n1,,3 Is plo looks like his-
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationTEACHER NOTES MATH NSPIRED
Naural Logarihm Mah Objecives Sudens will undersand he definiion of he naural logarihm funcion in erms of a definie inegral. Sudens will be able o use his definiion o relae he value of he naural logarihm
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationThe Contradiction within Equations of Motion with Constant Acceleration
The Conradicion wihin Equaions of Moion wih Consan Acceleraion Louai Hassan Elzein Basheir (Daed: July 7, 0 This paper is prepared o demonsrae he violaion of rules of mahemaics in he algebraic derivaion
More informationEXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN DIFFERENCE-DIFFERENTIAL EQUATIONS
Elecronic Journal of Differenial Equaions, Vol. 29(29), No. 49, pp. 2. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN
More informationLinear Dynamic Models
Linear Dnamic Models and Forecasing Reference aricle: Ineracions beween he muliplier analsis and he principle of acceleraion Ouline. The sae space ssem as an approach o working wih ssems of difference
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationf(t) dt, x > 0, is the best value and it is the norm of the
MATEMATIQKI VESNIK 66, 1 (214), 19 32 March 214 originalni nauqni rad research aer GENERALIZED HAUSDORFF OPERATORS ON WEIGHTED HERZ SPACES Kuang Jichang Absrac. In his aer, we inroduce new generalized
More informationThe Bloch Space of Analytic functions
Inernaional OPEN ACCESS Jornal O Modern Engineering Research (IJMER) The Bloch Space o Analyic ncions S Nagendra, Pro E Keshava Reddy Deparmen o Mahemaics, Governmen Degree College, Pormamilla Deparmen
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationAverage Number of Lattice Points in a Disk
Average Number of Laice Poins in a Disk Sujay Jayakar Rober S. Sricharz Absrac The difference beween he number of laice poins in a disk of radius /π and he area of he disk /4π is equal o he error in he
More information